Structural, Vibrational, and Thermal Properties of Densified Silicates

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Dtd uut1,2018) 17, August (Dated: .Bauchy M. l- h 3 ihx03) h iuae ytmi opsdo = N of composed is system simulated system The NS2 so-called x=0.33). the with to (close chosen been has system tt r rsne.Fnly eto umrzsthese summarizes section V liquid In section results. the Finally, topo- of presented. glass. structural, results are the state structural report and of we thermodynamics results III, IV, vibrational section and has In logical that methodology used. and been model numerical the the present allowing thus compressibility. proposed, isothermal an the is Eventually, of model density. computation state to behavior according to of the sensitive peak equation about very Boson trends the some be of report to we found Vibra- and also pressure the changes. are that anomalous properties and shows tional order occurs range environment tetrahedral medium silicon from transition octahedral increasing a to com- that with show particular properties Results the one density. study in and focus (NS2) and We position glassy silicate. densified sodium of liquid properties vibrationa structural, thermodynamics the and of description systematic a ing so performed of been has evolution pressure the far. to of according study system systematic the no knowledge, our eietlrslso tutr.Vibrational structure. on results perimental workers for pathways atoms diffusion sodium preferential and inhomogeneities ntoMlclrDnmc simulations Dynamics Molecular classical initio scale performed experi- been large recently to have superomputers, simulations compared Using successfully data. and mental studied been also elastic sjs etoe,(Na mentioned, just As h ril sognzda olw.I eto I we II, section In follows. as organized is article The allow- simulation Dynamics Molecular here present We n h iudsae ytmtcinvesti- systematic A state. liquid the and 22 oimslct N2 r investigated are (NS2) silicate sodium d ie lse r bevd uha the as such observed, are glasses sified repolymerization a observe We formed. 17–19 edces fisitniy ial,we Finally, intensity. its of decrease he ssibility. emdu ag re r observed, are order range medium he rpriso h ls tabetpesr have pressure ambient at glass the of properties oa caerlslcnenvironment, silicon octahedral an to it thl aiu codn to according maximum half at width yteBrhMrahnequation Birch-Murnaghan the by d ,722PrsCdx0,France 05, Cedex Paris 75252 u, I IUAINDETAILS SIMULATION II. eapaac ftrefl coor- three-fold of appearance he aesonavr odareetwt ex- with agreement good very a shown have 10–16 iuain rmCrakadco- and Cormack from Simulations . iiae nihsfrom insights : silicates d 2 O) x (SiO - 2 24–26 ) 1 − 23 x swl sab as well as , oee,to However, . ihx=0.30 with 11,20,21 and l 2

3000 atoms (700 Si, 1700 O and 600 Na), placed in a cubic box of various lengths L to study diﬀerent densities (from 1.5 g/cm3 to 5.4 g/cm3). To do so, all the simulations were run in the canonical ensemble (NVT). The room 5 temperature density27 of 2.466 g/cm3 is obtained with L=34.43A.˚ A computed pressure P = -1.6 GPa is found in the glass at this density. To take into account the oxidation state of atoms18, 4 partial charges are used for the Coulomb interaction, while the short-range Buckingham potential is of the -3 form : ρ = 4.5 g cm 3

r Cij (r)

Vij (r)= Aij exp(− ) − 6 (1) T ̺ij r g -3 ρ = 3.5 g cm where Aij , ̺ij and Cij are parameters which have been 2 fitted by Teter18. Usually, the Buckingham potential can induce spurious effects at high temperature (as V(r) can -3 go to negative infinity when r is close to zero, which ρ = 2.5 g cm leads to a collapse of the interacting atoms28. As de- 1 17 nij scribed in , a repulsive term Bij /r was introduced at short distance in order for the potential energy and its derivative to be continuous at r0 to avoid this issue. This potential has been extensively used by Cormack 0 et al.17,18 and has revealed a very good description of the 2 4 6 8 glass at room density for various compositions. The ef- r (Å) fect of pressure on such systems using classical Molecular Dynamics has been considered29 only at high temperature for the NS4 silicate by using a Born-Mayer interaction potential fairly similar to the one that is presently FIG. 1: (Color online) Total radial correlation function of MD used. While, at ambient pressure, the use of a Coulomb modeled sodium silicate glasses for increasing densities and comparison with neutron diffraction studies (white rounds) interaction with fixed partial charges is supported by the 36 ionic character of the interactions and the absence of from the work of Wright et al. (Neutron diffraction data). charge transfer, one may wonder to what extent fixed charges can be still considered with increasing pressure. 6 While we are not aware of any report of densified sili- 6000 K during 10 steps (2ns). Each melt was then con- cates, a recent ab initio Molecular Dynamics study (in tinuously cooled down to the selected temperature (from which electrons and charge transfer are explicitly com- 300 K to 4000 K) using a cooling rate of 10 K/ps. puted) on an oxide network-forming glass under high pressure30,31 has not shown any deformations of the elec- III. GLASS tronic cloud that would be significant enough for ambient pressure pseudopotentials to be modified. The mentioned example30,31, the consistency of the presently reported A. Real space properties results and the fact that the present potential was successfully used to reproduce a diffusion anomaly of O and 1. Total radial correlation functions Si atoms with increasing density16,32, also observed in 33 34 pure silica or water , suggest that a certain degree of The total correlation functions gT(r) for increasing confidence can be expected. densities are shown in Fig. 1. To check the validity of Classical Molecular Dynamics simulations were per- the simulated glass, comparison with experimental data formed using the DLPOLY package35. The equations (neutron diffraction from the work of Wright et al.36) of motion were integrated with the Verlet-Leapfrog algo- at room pressure was made. We recover the same level rithm, using a timestep of 2.0 fs. Coulomb interactions of agreement than in previous studies17,19. However, we were evaluated by the Ewald summation method with a notice an increased structured system with main peaks cutoff of 12.0 A.˚ The short-range interaction cutoff was being sharper as compared to experiments. This com- chosen at 8.0 A.˚ As mentioned, the simulations were parison has also been done by Cormack17. Using the run in the canonical ensemble (NVT) with a Berendsen same potential, a better agreement has been observed thermostat. by broadening the total correlation functions.37 The po- For each density, the system was first equilibrated at sition of the first Si-O peak is well reproduced, but is 3 found to be sharper than in experiments. The position 3. Coordination numbers of the second O-O peak is also well reproduced, suggesting a realistic O-Si-O angle in simulation. On the other In pure silica, the network in fully connected and the ˚ hand, simulation produces a peak at 3.1A arising from coordination number CN of Si and O atoms are found Si-Si correlations (see below) which is not present in ex- to be 4 and 2, in agreement with the stoichiometry of periments but merged with other contributions in the the glass (CN N = CN N ). This is not the case ˚ Si Si O O region 3-4A. It means that the inter-tetrahedral angle in sodium silicates since Na atoms create NBOs, thus Si-O-Si may be underestimated with respect to experi- disrupting the network. ments. This angle turns out to be highly sensitive to The distributions of IV, V and VI-fold coordinated sil- the employed potential. A detailed discussion about the icon atoms (SiIV, SiV and SiVI) can be obtained by enu- ability of the different potentials to reproduce the Si-O-Si 19 merating the number of oxygen neighbors in the first co- angle can be found in . ordination shell of each silicon atom. These populations As density increases, the first Si-O peak does not show are shown in Fig. 3a for each CN. The fraction of tetra- any shift in position but becomes broader, suggesting hedral SiIV atoms starts to drop from ̺ = 2.7 g/cm3 (P an increased disorder in the network, manifested by in- ≃ 1 GPa). At the same density, the fraction of SiV atoms creased coordination numbers (integral of the first peak). grows and reaches a maximum around ̺ = 4.0 g/cm3 (P As observed on the partial gi(r) distributions (see below), ≃ 28 GPa) prior to a continuous decrease as density in- the second peak is shifted to lower r and becomes broader. creases. The fraction of octahedral SiVI atoms increases from ̺ = 3.1 g/cm3 (P = 5 GPa) and this basic structure becomes predominant at high density. These trends are 2. Partial radial correlation functions rather usual in densified silicates. In amorphous silica, simulations from Tse38 predicted the increase of the Si CN to 5 at 15 GPa and up to 6 at 20 GPa. That trend The partial radial correlation functions gi(r) have been 39 computed from the pair correlation functions g (r) : was confirmed by simulations from Horbach . The ap- ij pearance of SiV and SiVI in densified sodium silicate has been confirmed experimentally using NMR.5,7 n 1 The environment of oxygen atoms has been analyzed g (r)= g (2) in the same fashion, i.e. by enumerating the number of i n ij Xj=1 silicon atoms in the first coordination shell of each oxygen atom. Here, Na atoms are not taken into account, We have split the analysis according to BO and NBO. this in order to distinguish BO from NBO and thus to These functions are shown in Fig. 2 for increasing den- split the Si CN analysis from the one involving the Qn I sities. While the position of the first peak in gSi (Si-O speciation which will be detailed below. Thus, O refers correlations) does not show any significant change, an in- to the oxygen atoms that are surrounded by only one sili- crease in the shoulder on the lower r side of the second con atom (i.e. NBO atoms). At low density, the fraction peak (Si-Si correlations) is observed as density increases, of NBO can be determined by x, the amount of soda, suggesting that the Si-O-Si angle decreases. As men- as each sodium atom creates one NBO. The fraction of tioned previously, the environment of the BO and NBO NBO fNBO is thus given by fNBO = NNBO/NO = 2x/(2- are studied separately using gBO and gNBO. For both, x). At x = 0.3, fNBO ≈ 0.35, which is consistent with the position of the first peak (O-Si correlations) remains simulation results in Fig. 3b. The fraction of OI drops the same, but important changes take place with density for densities larger than ̺ = 2.6 g/cm3. At this density, change for the second-neighbor correlation. The second the fraction of OII starts to increase, reaches a maximum peak (O-O correlations) is shifted to lower r and the dis- at ̺ = 3.8 g/cm3 and decreases at higher density. The tribution becomes broader. In the gNBO partial correla- present findings clearly indicate a repolymerization of the tion function, one notices the growth of a peak (at 2.4A˚ network through the creation of density induced Si-BO- for ̺ = 3.5 g/cm3) which contributes only to a shoulder Si connections at the expense of Si-NBO ones. They of the main peak at 2.6A˚ for ̺ = 2.5 g/cm3. This also are consistent with the decrease of the fraction of NBO suggests that densification affects the O-Si-O and Si-O- found experimentally from NMR in densified silicates6. Si angles rather than the Si-O distance between nearest As a consequence, the model of the network modifier Na neighbors. Finally, the first peak of gNa(r), associated atom simply given by stoichiometry (one Na atom involv- with Na-O correlations, is shifted to lower r. The de- ing the appearance of one NBO atom) does not remain crease in the Na-O distance with pressure has also been valid for ̺> 3 g/cm3. Ultimately, OIII are found at high observed using NMR by Lee6. The Na-centered pair dis- density and their fraction grows up to nearly 50% at ̺ tribution functions are highly sensitive to density change = 5.5 g/cm3. Note that 3-fold O atoms (termed triclus- and this is not surprising as it involves non-directional ters) have already been found both in experiments and bonds. However, we notice that the increase of density in simulations, for example in aluminosilicate glasses40. leads to a better defined first peak whose height increases At ̺ = 5.5 g/cm3, the fraction of NBOs is very low with the density. (≃ 3%) so that the Si/O network can be considered as 4

3 4

-3 ρ = 4.5 g cm 3 2 (r) (r) ρ -3

Si = 3.5 g cm Na g g 2 -3 ρ = 2.5 g cm 1 1

0 0 2 4 6 8 2 4 6 8 r (Å) r (Å) 5 5

4 4

3 3 (r) (r) BO NBO g

2 g 2

1 1

0 0 2 4 6 8 2 4 6 8 r (Å) r (Å)

FIG. 2: (Color online) Partial radial correlation function gSi(r), gNa(r), gBO(r) and gNBO(r) at different selected densities ̺ = 2.5, 3.5, 4.5 g/cm3. being fully connected as in pure silica, thus allowing us 4. Qn populations to check the agreement between the stoichiometry of the system (SiO2 43) and the computed coordination num- . As mentioned earlier, changes in the glass network can bers. On average, we find CN = 5.90 and CN = 2.43 Si O also be characterized by the Qn distribution analysis. We so that the stoichiometry of the glass is satisfied (CN N Si Si remind that a Qn silicon atom is defined as an atom ≈ CN N ). O O linked with n bridging oxygen atoms. Defining BO and NBO at high density needs a careful analysis since 3-fold coordinated oxygen atoms can be found. NBOs are here thus defined as oxygen atoms connected to only one Si. BOs are defined as oxygen atoms that are not NBOs. These results show that the network undergoes strong At ambient pressure, the Qn distribution usually range topological changes as density increases. The initial from a full Q4 (the silica network) to Q0 network, the or- tetrahedral silicon environment becomes octahedral at thosilicate glass, depending on the amount of soda x. At high density, consistently with the decrease of the O-Si-O ̺ = 2.5 g/cm3, the Q0,Q5 and Q6 populations were found angle (see below). On the other hand, a transition from to be negligible (less than 0.1% in each case), Q1,2,3,4 2-fold to 3-fold coordinated oxygen atoms is observed, populations from simulation being given in Table I and which is once again consistent with the decrease of the compared with results from a previous simulation17, with Si-BO-Si angle (see below). results of NMR studies41 and with results17 from a ran- 5

dom model proposed by Lacy42. First, we notice that our findings differ slightly with those obtained by Cormack17 using the same potential. The origin may be due to 1 IV VI the fact that the system has a different thermal history Si Si (the cooling methodology is slightly different although 0.8 the cooling rate is the same). However, both simulations are consistent with the random model. Differences with 0.6 V Si experimental data are important and these shifts have 23 0.4 been found even in very large-scale simulations . They Fraction have been attributed to the fact that the high cooling rate 0.2 used in simulations induces a structure with a higher ef- fective temperature43 so that the simulated Qn statistics 0 is the one of a high temperature frozen liquid. 2 3-3 4 5 (a) ρ (g cm )

n 0.8 II TABLE I: Proportion of Q populations at room density. O n Q Present MD MD Cormack17 NMR41 Random model42 0.6 Q1 1.288 1.857 0.000 2.985 Q2 18.598 15.571 4.776 16.716 I 3 0.4 O (NBO) III Q 44.067 49.000 68.358 41.493 4 Fraction O Q 35.908 33.571 26.567 37.910 0.2

0 (b) 2 3ρ -3 4 5 When density changes, we observe that the changes in (g cm ) Qn populations are correlated with the change in O and Si coordination numbers (Fig. 3), i.e. they take place only for ̺ > 3 g/cm3.Qn populations do not show any signiﬁcant changes at low density. At ̺ = 2.5 g/cm3, the FIG. 3: (Color online) Distribution of IV, V and VI-fold coor- Q5 proportion starts to increase however and reaches a dinated silicon atoms (a) and of I, II and III-fold coordinated maximum at ̺ = 4.0 g cm3. Again we notice a clear cor- oxygen atoms (b) with respect to density. Sodium atoms are relation between the Q5 population and the proportion not taken into account in the enumeration of the neighbors, V 6 I of Si . The Q proportion only starts to increase from so that O refer to NBOs. ̺ = 3.1 g /cm3, thus showing a behavior similar to the one of the proportion of SiVI.

5. Bond-angle distributions 0.8 We now focus on the bond-angle distributions (BAD) 6 Q and their variations with density, which have been 0.6 shown to be extremely sensitive in other tetrahedral systems38,44. Even at ambient pressure, it allows to un- 3 Q derstand how the basic structures of the glass connect to 0.4 each other. The O-Si-O BAD, which characterizes the Si 4 fraction tetrahedrons, is shown in Fig. 5a for three selected den- n Q 3 Q 5 sities (̺ = 2.5, 3.5 and 4.5 g/cm ). As expected at the 2 Q 0.2 Q lowest density (̺ = 2.5 g/cm3), the distribution is sharp and the average O-Si-O angle is close to the ideal 109.5◦ tetrahedral angle (see also Fig. 6a). At intermediate den- 0 sities however (̺ ≈ 3.5 g/cm3), the O-Si-O angle displays 2 3 4 5 ρ -3 now a bimodal distribution with a peak still located at (g cm ) 109◦, reminiscent of the initial tetrahedral structure, and a growing peak at 90◦. The latter corresponds to the an-

n gle that is expected for an octahedral environment. An FIG. 4: (Color online) Distribution of Q populations with additional signature for this environment is provided by increasing density. ◦ the contribution at 180 which appears for larger densities (Fig. 5a) and grows with ̺. At high density (̺ 6

O-Si-O ρ 3 1.5 = 2.5 g/cm 150 Si-BO-Si 1 3 4.5 g/cm 3 140 BAD 3.5 g/cm 0.5 130

> (degree) Si-NBO-Na

0 θ

(a) 60 80 100 120 140 160 180 < 120 Si-BO-Si 3 0.6 ρ = 2.5 g/cm 110 O-Si-O 3 3 3.5 g/cm 0.4 4.5 g/cm (a) 2 3 4 5 BAD 0.2 30 0 25 (b) 60 80 100 120 140 160 180 Si-NBO-Na 3 Si-NBO-Na 4.5 g/cm 20 3 0.4 3.5 g/cm ρ 3 = 2.5 g/cm (degree) θ 15

σ Si-BO-Si

BAD 0.2 10 O-Si-O 0 (c) 60 80 100 120 140 160 180 (b) 2 3 4 5 -3 Angle (degree) ρ (g cm )

FIG. 5: (Color online) O-Si-O (a), Si-BO-Si (b) and Si-NBO- FIG. 6: (Color online) First (a) and second (b) moment of the Na (c) bond angle distributions at ̺ = 2.5, 3.5 and 4.5 g O-Si-O, Si-BO-Si and Si-NBO-Na bond angle distributions. − cm 3.

4 GeO2 . = 4.5 g/cm3, all silicon atoms display an octahedral en- On the contrary, the Si-NBO-Na BAD does not show vironment with a single peak at 90◦ (and the vanishing any significant change with density. of the tetrahedral peak at 109◦) and the contribution at 180◦. The second moment of the O-Si-O BAD is shown in Fig. 6b and grows from 5◦ at ordinary density (̺ = 2.5 6. Orientational parameter g/cm3) up to more than 30◦ at high density, suggesting the appearance of a pressure induced disorder manifested An interesting means to analyze the tetrahedral to oc- by an increased angular excursion around a mean value. tahedral conversion in liquids and glasses is provided by The Si-BO-Si angle characterizes the way two adjacent the orientational order parameter q (introduced by Chau 48 34 silicon tetrahedrons are connected. At ordinary density, and Hardwick and rescaled in ), which quantifies the the angle shows a broad distribution between 120◦ and extent to which a molecule and its four nearest neighbors 180◦ and centered at 153◦ (see also Fig. 6a), compared to adopt a tetrahedral arrangement. It is defined by : the 142◦ experimental value from NMR45,46. This difference was also observed by previous simulations, as re- 3 4 19 3 1 viewed in and is consistent with the over estimated q =1 − h (cos θ + )i (3) 8 ijk 3 value of the Si-Si distance. Like the O-Si-O angle, the Xi=1 k=Xj+1 Si-BO-Si angle displays a bimodal distribution as density increases. The BAD shows a second peak close to where θijk is the angle formed by the central Si atom 100◦ at high density (̺ = 4.5 g/cm3). This contribution i and its oxygen nearest neighbors j and k, the brack- is absent at intermediate densities (̺ = 3.5 g/cm3, see ets representing an average over the central Si atoms i Fig. 5b) but the trend with ̺ is clearly correlated with and over the time. This parameter is normalized so that the population of OIII (see Fig. 3b and Fig. 6a). The its average varies between 0 (randomly arranged bonds) decrease of the average value of the Si-BO-Si angle has and 1 (perfect tetrahedral network). It has been used also been observed experimentally in silicates47 and for for the analysis of diffusivity anomalies in relationship 7

at ̺ = 4.5 g/cm3, suggesting once again the appearance of a pressure induced disorder. 8 -3 ρ = 2.5 g cm B. Reciprocal space properties 6 To investigate the structure of the glass on intermediate length scales, the neutron structure factor has been 4 ρ -3 = 3.5 g cm computed. The partial structure factors have been ﬁrst ρ -3 calculated from the pair distribution functions gij (r) :

q distribution = 4.5 g cm 2 R 2 sin(Qr) S (Q)=1+̺0 4πr (g (r)−1) F (r) dr (4) ij Z ij L 0 0 Qr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q where Q is the scattering vector, ̺0 is the average atom number density and R is the maximum value of the integration in real space (here R = 15A).˚ The FL(r) = FIG. 7: (Color online) q factor distributions for increasing sin(πr/R)/(πr/R) term is a Lortch-type window function densities. used to reduce the effect of the finite cutoff of r in the integration50. As discussed in51, the use of this function reduces the ripples at low Q but induces a broadening of the structure factor peaks. The total neutron structure 0.75 factor can then be evaluated from the partial structure factors following : 0.12 0.7 0.1 n n −1 0.65 SN (Q) = ( cicj bibj ) cicj bibj Sij (Q) (5) 0.08 i,jX=1 i,jX=1 q

0.6 σ 0.06 where ci is the fraction of i atoms (Si, O or Na) and bi 0.55 is the neutron scattering length of the species (given by 0.04 5.803, 4.1491 and 3.63 fm for oxygen, silicon and sodium 0.5 atoms respectively52). 0.02 0.45 1.5 2 2.5 3 3.5 4 4.5 5 -3 1. Neutron structure factor ρ (g cm )

The total neutron structure factor SN for different in- FIG. 8: (Color online) First (left axis) and second (rigt axis) creasing densities are shown on Fig. 9. The room den- moment of the q factor distributions for increasing densities. sity structure factor is compared both with the neutron diffraction results from Wright et al.36 and the simulated NS2 glass from Horbach et al.14, using an alternative with tetrahedral to octahedral changes in liquid water34, (BKS) potential. We note that the agreement between silica33 and germania49. simulation and experiment is good. The agreement of the first peak with experiment is discussed in details in Fig. 7 shows the calculated distribution of q values 3 the FSDP section below. The second peak position is well for 3 selected densities (̺ = 2.5, 3.5 and 4.5 g/cm ). − − reproduced (3.0A˚ 1 experimentally, compared to 3.0A˚ 1 At room density, the distribution exhibits only a sharp ˚−1 peak close to q = 0.7 (see Fig. 8), which corresponds from the present potential and 2.9A from the BKS po- to near-perfect tetrahedral order, as also found for silica tential). The third peak position is also very well repro- ˚−1 ˚−1 at ambient pressure33. At ̺ = 3.5 g/cm3, the observed duced (5.4A experimentally, compared to 5.3A from −1 bimodal distribution suggests the existence of both tetra- the present potential and 5.2A˚ from the BKS poten- hedral Si (high q peak) and higher coordinated Si (low q tial). peak with q < 0.6). At ̺ = 4.5 g/cm3, the tetrahedral As observed on Fig. 9, the density mainly influences contribution has vanished. The second moment of the the low wave vector part of the structure factors, which distributions according to density is shown in Fig. 8. It suggests that the main effects of density do not apply at characterizes the orientational disorder around central Si short length scales. The shape in the high Q limit (at −1 3 atoms, with σq going from 0.02 at low density up to 0.1 Q>10A˚ is nearly unchanged for ̺ 6 3.5 g/cm . All 8

3.5 -3 3 6 ρ = 4.5 g cm -3 ρ = 4.5 g cm 4 ρ -3 2.5 (Q) = 3.5 g cm -3 -3 ρ = 3.5 g cm Si-O 2 ρ = 2.5 g cm 2 S (Q) -3 N ρ = 3.0 g cm 0 S 1.5 -3 2 4 6 8 10 12 14 ρ = 2.5 g cm 1 -3 6 ρ = 4.5 g cm -3 0.5 (Q) 4 ρ = 3.5 g cm

2 4 6 8 10 12 14 O-O -3 -1 S 2 ρ = 2.5 g cm Q (Å ) 0 2 4 6 8 10 12 14 FIG. 9: (Color online) Neutron structure factors for increas- -3 ing densities. Neutron diﬀraction results from Wright et 6 ρ = 4.5 g cm al.36 (open circles) and simulation results from Horbach at (Q) -3 al.14 (dotted line) are shown for comparison. Examples of 4 ρ = 3.5 g cm

Lorentzian fits of the FSDP are displayed in orange. Na-O -3 S 2 ρ = 2.5 g cm 0 2 4 6 8 10 12 14 peaks are shifted to higher wave vector (lower r) as den- -1 sity increases, which is linked to the compaction of the Q (Å ) network. Interestingly, the second moments of the different peaks do not show the same behavior with density. The so-called first sharp diffraction peak (FSDP) at very low Q becomes broader and less intense with increasing FIG. 10: (Color online) Partial structure factors Si-O, O-O density, as discussed below. The main peak around 3A˚−1 and Na-O for increasing densities. becomes narrower as density increases, whereas the third −1 one (around 5A˚ ) becomes broader. These trends can − − 3A˚ 1) and of the second main peak (≃ 5A˚ 1) with den- be analyzed in more details from the partial structure ˚−1 factors (see below). The shape of the other peaks are sity. Indeed, the peak at 3A in O-O and Na-O par- almost unaffected by density. tial structure factors becomes narrower as density increases, an effect which is related to the increased structural medium-range order at high density. These peaks contribute the most of the second peaks of the total struc- 2. Partial structure factors ture factor. On the other hand, the main contribution for the peak at 5A˚−1 of the total structure factor arises Fig. 10 shows the decomposition of the total structure from the second Si-O partial structure factor peak, which factor into contributions of different pair structure factors becomes broader as density increases. This apparent dis- SSi-O(Q), SO-O(Q) and SNa-O(Q) for different increasing order may be attributed to the appearance of coexisting densities. The partial SSi-Si(Q), SNa-Na(Q) and SSi-Na(Q) tetrahedral and octahedral Si-O environment as density decay the fastest and have thus not been displayed. At increases. normal density, the shape of these pair structure factors and the positions of the peaks are in excellent agreement with previously reported results from MD simulations53. 3. First sharp diffraction peak At room pressure, the partial SSi-O(Q) shows the most significant variations with Q both at short wave vector, FSDPs are not simply the first of the many peaks correlated to the medium-range order of the silicate net- of any diffraction pattern but display many anomalous work, and at long Q, correlated to the strong short-range behavior as a function of temperature, pressure and 54 Si-O order. As already observed in the total structure composition. Since the position of the FSDP QFSDP factor, most of the peaks are also shifted to higher Q is smaller than QP (the position of the principal peak of (lower r) as density increases. the structure factor, associated to the nearest-neighbor The decomposition of the total structure factor can distance), the FSDP corresponds to structural correla- serve to understand the behavior of the main peak (≃ tions on a larger length scale. This feature has been ob- 9

Total 4 Si-O 0.8 2.4 Total Si-O 3.5 Total d (Å)

FSDP 0.4 O-O I O-O 3 O-Na 2.2 2 3 4 0 -3 ρ (g cm ) O-Na ) -1 O-Na (a) 1.5 2 2.5 3 3.5 4 4.5 5 (Å 2 7 Total 1.6

FSDP O-O 0.8 6 O-Na Q

L (Å) 5 1.4 0.6 4 2 3 4-3 5 1.8 ρ (g cm ) Si-O 0.4 1.2 Si-O

1 (Q) FSDP FWHM (Å)

(Q) FSDP FWHM (Å) 0.2 ij 1.6 N S O-O S 0 0.8 1.5 2 2.5 3 3.5 4 (b) 1.5 2 2.5 3 3.5 4 4.5 5 -3 -3 ρ (g cm ) ρ (g cm )

FIG. 11: (Color online) FSDP position of the total Neutron FIG. 12: (Color online) (a) Intensity of the FSDP of the total structure factor and positions of each relevant partial struc- and partial structure factors. (b) FSDP FWHM of the total ture factors FSDP. The insert shows the associated character- and partial structure factors. The insert shows the correlation istic distance d = 2π/QFSDP of the total and partial structure length L = 2π/FWHM. factors.

density. Fig. 11 and 12 show the position QFSDP, the 55,56 57 served both in covalent and ionic amorphous sys- intensity IFSDP and the FWHM of the FSDP. The com- tem. In ionic systems, this medium range order has been puted FSDP position at room density (1.85A˚−1) is found associated to the forced separation between cations be- to be in very good agreement with the one obtained from cause of their mutual Coulomb repulsion, thus producing experiment (1.83A˚−1)64 and with the one from the MD a prepeak in the cation-cation structure factor58. Pre- work by Corrales et al. (1.77A˚−1)51. Except at very low peaks can also arise from size effects of the atoms of the density (̺ < 2.5 g/cm3, negative pressure domain), the network59. However, the network formation itself can FSDP position increases with density while its intensity have a major role since the FSDP is also observed in the decreases. This trend is consistent with X-ray diffraction monoatomic tetravalent systems a-Si and a-Ge60,61. The results from Benmore in densified silica.65 Interestingly, FSDP origin is now usually explained by using a void- the FHWM of the FSDP exhibits a density window be- based model54,62 in which ordering of interstitial voids tween 2.3 and 3.3 g/cm3 with a minimum found at ̺ = occurs in the structure. 2.7 g/cm3. The FSDPs we obtained from simulations were further Coming back to the real space correlations, the FSDP studied by fitting them with Lorentzian functions (exam- peak position QFSDP is usually related to a character- ples of fitted functions can be seen on Fig. 9). This choice istic repeat distance d = 2π/QFSDP and the FWHM is supported by the fact that the experimental results in to a correlation length L = 2π/FWHM, sometimes neutron scattering factor of silica can be better fitted also called ’coherence length’, due to atomic density with a Lorentzian function than with a Gaussian one63. fluctuations66,67. The effect of irradiation36,68, water It should be noted that the fit has been done on the low content67,69 and alkali content51 on the FSDP have been Q part of the FSDP to avoid the contribution of the fol- studied, leading to the idea that a depolymerization of lowing peaks. This allows to track precisely intensity, the network (a decrease of the atomic order) is associ- position and full-width at half maximum (FWHM) with ated to a decrease of the intensity of the FSDP and a 10 decrease in the characteristic distance d. A global understanding of the correlation length L is lacking since it has been found to decrease with increasing potassium amount in silica network, to increase with increasing From VAF From DM lithium amount and to show a maximum in sodium sili- 6 Zotov simulation cate when x = 0.2051. It seems therefore highly system dependent.

) 4 The inserts of Fig. 11 and 12b show the computed ω Na characteristic distance d and characteristic correlation g( BO length L as a function of density. It can be observed NBO that the characteristic distance d decreases with density, 2 Si which suggests a decrease of the medium range order (MRO). On the contrary, the correlation length L does not follow a general behavior with density since it shows 0 3 0 5 10 15 20 25 30 35 40 a density window between 2.3 and 3.3 g/cm with a max- ω (THz) imum at 2.7 g/cm3. We notice that the density of the maximum of L corresponds to the density of the begin- ning of the growth of the SiV fraction (see Fig. 3a). FIG. 13: (Color online) Vibrational density of states at room pressure computed from the Fourier transform of the velocity autocorrelation function (VAF) and from the diagonalization of the dynamical matrix (DM). The results are compared with the one from the simulation of Zotov et al.70 using a different 4. Contributions to the FSDP potential. The partial VDOS for Si, BO, NBO and Na are also shown. Even though it can be noticed from Fig. 10 that all partial structure factors show a FSDP, they do not con- C. Vibrational properties tribute to the FSDP of the total structure factor at the same level. To understand the behavior of the FSDP, the position, intensity and FWHM of the FSDPs of each The nature of the vibrational excitations of silicate partial structure factor Sij (Q) have been computed. glasses has so far remained a challenging issue. As contrary to crystals, the lack of long-range structural or- Fig. 11 and 12a show the position and the inten- der in amorphous solids strongly affects their vibrational sity of the FSDPs according to density. At low den- dynamics. The appearance of an excess of vibrational sity (̺ < 2.7g/cm3), the main contribution to the total modes over the Debye level at terahertz frequencies, the FSDP clearly comes from the Si-O FSDP, since their po- so-called Boson peak (BP), is one of the special features sition and intensity are similar and show the same trend. exhibited by glasses. However, at larger densities, the partial FSDPs positions show a maximum (around ̺ = 3.1g/cm3) whereas the total FSDP continuously increases. These maximums 1. Vibrational density of states correspond to minimums of the characteristic repeat distance d and we notice that they occur at the density at which the SiVI fraction starts to grow. This shows that The vibrational density of states (VDOS) g(ω) can be the total FSDP is not a simple superposition of the par- computed in two different ways. Starting from a relaxed tial FSDP. The shift of the total FSDP to higher Q at glass (via energy minimization or cooling to 0K), one high density can mainly be explained by the increased can compute the dynamical matrix (DM) by evaluating −1 contribution of the main peak of the SO-O (at 3A˚ ) the second derivative of the total energy with respect to 71 whose intensity grows with the density. small atomic displacements . The diagonalization of the DM provides the eigenvalues, i.e. the frequency of each Even though the link between the total and the par- normal vibrational mode. Another way is to compute the tial FSDPs is not simple, it is interesting to notice that Fourier transform of the velocity autocorrelation function each partial FSDPs show a minimum of their FWHMs (VAF) : according to the density (see Fig. 12b). Si-O and O- Na partial FSDPs FWHM reach their minimums around 3 2.7g/cm , corresponding to the minimum of the to- N ∞ 1 tal FSDP FWHM. The O-O partial FSDP shows the g(ω)= mj < vj (t)vj (0) > exp(iωt) dt 3 Nk T Z−∞ sharpest minimum of its FWHM around 3.2g/cm (i.e. B Xj=1 at larger density than for the total FSDP) but does not (6) contribute a lot to the total FSDP due to its low intensity where N is the number of atoms, mj is the mass of an (see Fig. 12a). atom j, ω is the frequency and vj (t) is the velocity of 11

-3 -3 ρ = 2.5 g cm ρ = 2.5 g cm 6 -3 ρ = 3.5 g cm 1 -3 -3 ρ = 3.5 g cm ρ = 4.5 g cm 2 ) 4 ω )/ ω ω g(

g( -3 ρ = 4.5 g cm 2 0.1

0 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 ω (THz) ω (THz)

FIG. 14: (Color online) Vibrational density of states, com- FIG. 15: (Color online) Boson peak visualization for different puted from the VAF, for different selected densities. selected densities. an atom j. It has been reported that both methods lead most at high frequency (27-37 THz, symmetric and anti- to quite similar VDOS in silica72. Although the DM ap- symmetric stretching modes) and at intermediate fre- proach is far more expensive computationally than the quency (22 THz, O-Si-O bending mode). BO atoms pre- VAF one, it should be noted that, looking at the eigen- dominant contributions occur at high frequency (29-37 vectors ei associated to each eigenvalue frequency ωi, one THz, symmetric stretching modes) and at low and inter- can get the details of the nature of each normal mode. mediate frequency (0-26 THz, Si-BO-Si bending mode One can, for example, compute the partial VDOS gα(ω) and symmetric stretching mode). The contribution of (α = Si, BO, NBO, Na) for each atom defined as : NBOs differs from the one of the BOs because of the decreased number of O-Si stretching modes and of the softening of the Si-NBO-Na bending mode as compared e 2 gα(ωi)= g(ωi) | j (ωi)| (7) to the Si-BO-Si mode. Most of the low frequency contri- Xj∈α bution comes from Na atoms (0-10 THz, Na-NBO low- energy stretching modes). e where j (ωi) are the 3-component real space eigenvectors Fig. 14 shows the vibrational density of states, scaled associated to the atoms α. to one, for different increasing densities. We observe Fig. 13 shows the VDOS, scaled to one, computed us- some trends which are similar to the ones observed ing the two previously described methods. We observe a in densified silica74 : decrease of the number of low- fair agreement between both methods, especially at low frequency modes, disappearance of the gap between in- and intermediate frequency. The difference at high fre- termediate and high frequency modes around 27 THz quency can be explained by the harmonic assumption on and broadening of the high-frequency peak. The low- which the DM method relies. The results are compared 70 frequency region, whose contribution mainly comes from with the one from the simulation of Zotov et al. . We Na atoms, is the most affected part of the VDOS, suggest- observe that the agreement is very poor even if some ing that the Na vibrational modes are strongly modified trends are similar (sharp peak at high frequency and ap- during densification. pearance of a new peak at low frequency increasing with As density increases, the low-frequency peak coming respect to the amount of sodium). We note that the sim- from Na-O bounds decreases in intensity and is shifted ulation from Zotov et al. uses a different potential (from to higher frequencies. The high frequency peak coming Vessal) involving both 2- and 3-body terms and that the from Si-O bounds becomes broader but does not show system is quite smaller (1080 atoms, as compared to 3000 any significant frequency shift. in the present simulation). Unfortunately, to our knowledge, no experimental VDOS is currently available for this composition. Using Eq. 7, the partial VDOS have been computed 2. Boson peak and are shown on Fig. 13. Note that O atoms have been split into BOs and NBOs. Relying on the vi- The origin of the BP in silica, still controversial, has brational analysis of silica72,73, one can interpret some been associated to the existence of local modes involving 75–77 features of the present VDOS. Si atoms contribute the rocking motions of distorted SiO4 tetrahedrons . The 12

3 10 T = 3000 K 100 2.5 15

2 10 10 1 Pressure (GPa)

(THz) 1.5 BP (arb. unit) 1

ω 5 BP

1 I

Pressure (GPa) 2 3 4 5 -3 Density ρ (g cm ) 0.5 0 T = 1500K 0.1 glass 0 1.5 2 2.5 3 3.5 4 4.5 5 15 20 25 30 35 -3 3 -1 ρ (g cm ) Volume (cm mol )

FIG. 16: (Color online) Boson peak position (left axis) and FIG. 17: (Color online) Isotherms for glass and liquid NS2. intensity (right axis) with respect to density. The curves are separated by 500K each. The inset shows the corresponding data in (̺, P) together with the BM ﬁts (solid lines). BP can be observed by looking at the excess VDOS over 2 the Debye law g(ω) − gD(ω) or at the quantity g(ω)/ω .

The latter quantity can be identiﬁed with the one-phonon ) scattering cross section as measured in neutron scatter- -1 77,78 ing experiments and is shown on Fig. 15 for three 0.1 T=3000K 12 (GPa selected densities. A pronounced peak can be observed T at each density, even if its intensity decreases with den- κ 11 sity. At room pressure, the BP is found to be located 10

at ωBP=1.3 THz, which is lower than the value found Bulk Modulus at P=0 (GPa) −1 79 T=1500K 1500 2000 2500 3000 experimentally (Raman) of 1.95 THz (65 cm ) . Temperature (K) The BP properties have been further analyzed by com- 0.01

) 2.6 -3 puting its position ωBP and its intensity IBP, quantities 2.4 (g cm 0

that are displayed on Fig. 16. It can been observed that ρ 2.2 IBP decreases with density, while ωBP increases with den- 1000 2000 3000 sity. Both of these two trends (decrease of the intensity Temperature (K) and increase of the frequency) have been observed ex- 2 2.5 3 3.5 4 4.5 5 5.5 Isothermal Compressibility -3 perimentally in many system, such as in pure silica80, in Density ρ (g cm ) 81 82 lithium silicate glass , in a Na2FeSi3O8 glass as well as in diﬀerent polymers83. FIG. 18: (Color online) Isothermal compressibility κT with respect to density in liquid NS2 for various temperatures rang- IV. LIQUID ing from 3000 to 1500K separated by 500K each. The curves are computed using the BM EOS. The inset shows the room A. Thermodynamics pressure density ̺0 change with temperature.

To evaluate the equation of state (EOS) of the liquid, used in geophysical studies (see for example89). many thermodynamics points have been computed (T, ̺, Fig. 17 shows the isotherms of the glass and the liquid P). The following range has been studied : 1.5 6 ̺ 6 5.5 3 in the (P, V) representation. The data have been fitted g/cm and 1500 6 T 6 3000 K, which correspond to the far from the critical region with the BM EOS, that has following pressure range : −2.23 6 P 6 150 GPa. In con- the following form : trast with previous works on molecular fluids84,85 and silica, where the data were fitted using a Van der Waals type EOS, the data of the current simulations were fitted with 3 ̺ 7/3 ̺ 5/3 3 ̺ 2/3 P = K(( ) − [ ] )(1 − (4 − K1)([ ] − 1)) a Birch-Murnaghan equation of state (BM EOS) that has 2 ̺0 ̺0 4 ̺0 a simpler form86,87. It has revealed to give reasonable fits (8) 88 in the case of a liquid densified germania and is widely where K is the bulk modulus at P=0, K1 = dK/dP at P 13

5 1 -3 ρ = 2.5 g cm IV Si VI 4 T = 2500K 0.8 Si

T = 2000K V 3 0.6 Si (r) T g 2 T = 1500K 0.4 Fraction T = 300K 1 0.2

0 0 2 4 6 8 1.5 2 2.5 3 3.5 4 4.5 5 r (Å) 5 -3 ρ = 4.5 g cm II T = 2500K 0.8 O 4 T = 2000K 0.6 3

(r) I T T = 1500K g O (NBO) III 2 0.4 O Fraction T = 300K 1 0.2

0 0 2 4 6 8 1.5 2 2.5 3 3.5 4 4.5 5 r (Å) -3 ρ (g cm )

FIG. 19: (Color online) Radial distribution function for in- FIG. 20: (Color online) Distribution of IV, V and VI-fold creasing temperatures. coordinated silicon atoms (a) and of I, II and III-fold coordinated oxygen atoms (b) with respect to density both at 300K (filled symbols) and 2000K (open symbols). Sodium atoms = 0 and ̺0 is zero-pressure density of the liquid. The fit are not taken into account in the enumeration of the neighbors, so that OI refer to NBOs. can be made with two parameters only (K and K1) since ̺0 can be accessed from the isothermal data displayed on Fig. 17. It can be seen that the data are very well fitted by the BM EOS along all the density and temperature icant shift as temperature increases. The only behavior range. that can be seen is a broadening of all peaks as tempera- In addition, the BM EOS allows to have access to the ture increases, which can be explained by the increasing bulk modulus K at P=0 and to the isothermal compress- disorder due to the increasing thermal energy. The same −1 trends are observed in pure silica91. ibility κT = ̺ (∂̺/∂P )T according to the density (plotted on Fig. 18). The observed behavior, enhanced com- Eventually, the influence of the temperature on the co- pressibility with falling density, is realistic, as well as the ordination numbers has been checked. The populations decrease of the bulk modulus at P = 0 with respect to of the different Si and O species according to the density the temperature (see the insert of Fig. 18). The results are plotted on Fig. 20 both at T = 300K and 2000K. The IV V VI are in good agreement with the only experimental data transition between Si into Si and Si is still clearly observed, although the transition occurs at a lower den- on liquid NS2 we are aware of (K = 13.4 GPa and κT = 0.075 GPa−1 at T=1500 K and P = 0)90. sity than in the glass (density shift of approximately 0.2 g/cm3). The same shift can be observed for the O species.

B. Structure V. CONCLUSION

Fig. 19 shows the total correlation function gT(r) for different increasing temperatures, both at low and high Our purpose in the present paper has been to provide a density. It can be observed that density seems to have systematic and extensive study of the properties of den- a more critical influence on structure than temperature. sified glassy and liquid NS2 sodium silicate. Indeed, the positions of the peaks do not show any signif- While bond distances remains nearly unchanged, pres- 14 sure has a strong effect on angles and coordination num- sure, it is well-known that changes in composition of the bers. A transition from tetrahedral to octahedral silicon glass (and especially in the amount of sodium atoms) in- environment is found. The fraction of NBOs decreases duce changes of the degree of polymerization of the glass. and OIII tricluster are observed at high density. The These competitive effects (depolymerization by sodium usual vibrational behavior is observed, i.e. the decrease atoms and repolymerization by the pressure) should be of the amount of low frequency modes, the increase of the addressed in the future for a better understanding of the frequency of the Boson peak and the decrease of its in- glass network properties. tensity under pressure. Expected anomalous effects are found in the medium range order (increase of the position of the FSDP and decrease of its intensity under pressure), but, more surprisingly, we observe a minimum Acknowledgments of the FWHM of the FSDP according to the density. Temperature is found to have only small effects on structure and the Birch-Murnaghan equation of state allows Warm thanks are due to M. Micoulaut for suggesting to reproduce the densification of the liquid at each tem- this study and providing advice at its various stages, to J. perature. C. Mauro for his help to compute the Dynamical Matrix Finally, it is worth mentioning that, as ambient pres- and to G. Mountjoy for a very stimulating discussion.

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